SAS中混沌控制
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摘要
开关到达系统作为一种典型的混合系统,会产生混沌现象。为抑制其混沌现象,相继产生设定缓冲器值的上限或下限方法、设定服务器连续注入时间上限方法、可控内部连通方法、时滞脉冲反馈控制方法等。时滞脉冲反馈控制是时滞反馈控制与脉冲控制的结合与改进,本文将该方法的控制范围扩展到每个边界来控制混沌(多边控制),分析了单边控制与多边控制的收敛性。接着假定开关时间大于0时,开关到达系统会产生损失情况。对该方法控制时间的选择加以修改,确保其不仅能控制混沌现象,还能降低系统损失率。主要结论与方法如下:
1、介绍开关到达系统,引入庞加莱截面来定义系统中不稳定周期轨,列举和对比针对该系统混沌现象的经典控制方法。
2、对开关到达系统实施多边控制(如N=3):提出控制方法;阐述控制时间的选择以及控制具体实施过程;数值验证控制目标为2-周期轨、3-周期轨时,该方法的控制效果,并且探测到其它周期轨的具体位置;分析该方法收敛性;证明控制系数矩阵K的存在性。
3、引入渐进收敛速度和迭代矩阵谱半径概念来分析单边与多边控制收敛性,讨论迭代矩阵谱半径与控制效果之间的关系。数值检验单边与多边控制的效果。
4、假定开关时间大于0时,系统会产生损失的情况。通过比较开关频率大小后,确定1-周期轨为控制目标。对控制时间加以改变来避免开关事件在实施控制时发生。数值验证不同情况下,有无实施多边控制效果的对比。
Switched arrival system is used as a kind of typical hybrid system, which can lead to chaotic behavior. In order to control its chaotic behavior, many methods are proposed in succession, such as upper and lower limits placed on the volumes of the buffers, changing the limited continuous processing time, controlled internal connections and delayed impulsive feedback control. Delayed impulsive feedback control is synthesis improvement of impulsive control and time-delayed feedback control. Chaos is controlled by implementing control at each boundary (multi-sided control) in this paper. Convergence of one-sided and multi-sided control methods is also analyzed. Then under the assumption of switched time being more than zero, switched arrival system leads to loss of production. Selection of control time is changed such that chaos is not just controlled, but also loss rate of the system is decreased. The main results and methods are as follows:
1 Switched arrival system is introduced. Poincare section is selected for defining unstable periodic orbits. The typical methods of chaos control are introduced.
2 Multi-sided control method is implemented in switched arrival system (such as N = 3). Control method is proposed; Selection of control time and implementation of control method are described; When control target are period-2 orbit and period-3 orbit, the effectiveness of method is numerically examined, and periodic orbits with other period are also detected. Convergence of the method is analyzed; Existence of control coefficient matrix K is proved.
3 Asymptotical convergence rate and spectral radius of iterative matrix are introduced to analyze convergence of one-sided control and multi-sided control. Relation between spectral radius of iterative matrix and effectiveness of the method is discussed. One-sided control and multi-sided control are numerically examined.
4 Under the assumption of switched time being more than zero, switched arrival system leads to loss of production. By comparing switched frequency, period-1 orbit is selected as control target. Control time is changed to avoid occurrence of switched event in the process of control. Loss rate of the system with or without implementing multi-sided control are compared.
1、介绍开关到达系统,引入庞加莱截面来定义系统中不稳定周期轨,列举和对比针对该系统混沌现象的经典控制方法。
2、对开关到达系统实施多边控制(如N=3):提出控制方法;阐述控制时间的选择以及控制具体实施过程;数值验证控制目标为2-周期轨、3-周期轨时,该方法的控制效果,并且探测到其它周期轨的具体位置;分析该方法收敛性;证明控制系数矩阵K的存在性。
3、引入渐进收敛速度和迭代矩阵谱半径概念来分析单边与多边控制收敛性,讨论迭代矩阵谱半径与控制效果之间的关系。数值检验单边与多边控制的效果。
4、假定开关时间大于0时,系统会产生损失的情况。通过比较开关频率大小后,确定1-周期轨为控制目标。对控制时间加以改变来避免开关事件在实施控制时发生。数值验证不同情况下,有无实施多边控制效果的对比。
Switched arrival system is used as a kind of typical hybrid system, which can lead to chaotic behavior. In order to control its chaotic behavior, many methods are proposed in succession, such as upper and lower limits placed on the volumes of the buffers, changing the limited continuous processing time, controlled internal connections and delayed impulsive feedback control. Delayed impulsive feedback control is synthesis improvement of impulsive control and time-delayed feedback control. Chaos is controlled by implementing control at each boundary (multi-sided control) in this paper. Convergence of one-sided and multi-sided control methods is also analyzed. Then under the assumption of switched time being more than zero, switched arrival system leads to loss of production. Selection of control time is changed such that chaos is not just controlled, but also loss rate of the system is decreased. The main results and methods are as follows:
1 Switched arrival system is introduced. Poincare section is selected for defining unstable periodic orbits. The typical methods of chaos control are introduced.
2 Multi-sided control method is implemented in switched arrival system (such as N = 3). Control method is proposed; Selection of control time and implementation of control method are described; When control target are period-2 orbit and period-3 orbit, the effectiveness of method is numerically examined, and periodic orbits with other period are also detected. Convergence of the method is analyzed; Existence of control coefficient matrix K is proved.
3 Asymptotical convergence rate and spectral radius of iterative matrix are introduced to analyze convergence of one-sided control and multi-sided control. Relation between spectral radius of iterative matrix and effectiveness of the method is discussed. One-sided control and multi-sided control are numerically examined.
4 Under the assumption of switched time being more than zero, switched arrival system leads to loss of production. By comparing switched frequency, period-1 orbit is selected as control target. Control time is changed to avoid occurrence of switched event in the process of control. Loss rate of the system with or without implementing multi-sided control are compared.
引文
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[2]Ruelle D,Takens F.On the nature of turbulene[J].Commun Math Physics,1971,20:167-192
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[4]May R M.Simple mathematical models with very complicated dynamics(J].Nature,1976261(5560):459-467
[5]Feigenbaum M J.Quantitative universality for a class of nonlinear transformation(J].Statistical Physics,1978,19:25-52
[6]Guckenheimer J,Holmes P J.Nonlinear oscillations,dynamical systems,and bifurcations of vector fields(M].Springer-Verlag:New York,Berlin,Heidelberg,1983
[7]刘秉正.非线性动力学与混沌基础[M].长春:东北师范大学出版社,1994
[8]Wiggins S.Introduction to applied nonlinear dynamical systems and chaos(M].Springer-Verlag:New York,Berlin,Heidelberg,1990
[9]马知恩,周义仓.常微分方程定性与稳定性方法[M].北京:科学出版社,2005
[10]张琪昌,王洪礼,竺致文,等.分又与混沌理论与应用[M].天津:天津大学出版,2005
[11]关新平,范正平,陈彩莲,华长春.混沌控制及其在保密通信中的应用[M].北京:国防工业出版社,2002
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[13]杨凌,刘曾荣.OGY方法的改进及证明[J].应用数学和力学,1998,19(1):127-130
[14]Romeiras F J,Grebogi C,Ott E,Dayawansa W P.Controlling chaotic dynamical systems [J].Physics Letters D,1992,58:165-169
[15]Pyragas K.Continuous control of chaos by self-controlling feedback(J].Physics Letters A,1992,170:421-428
[16]Pyragas K.Experimental control of chaos by delayed self-controlling feedback(J].Physics Letters A,1993,180:99-102
[17]Huberman B A.Dynamics of adaptive systems(J].IEEE Transactions on Circuits and Systems,1990,37(4):547-550
[18]Sinha S.Adaptive control in nonlinear dynamics.Physics Letters D,1990,43:118-128
[19]Utkin V I.Sliding modes and their application in variable structare systems(M].Moscow,Russia:Mir Publishers,1978
[20]Utkin V 1.Variables structure systems:present and future(J].Automation and Remote Control,1983,44(9):1105-1120
[21]高为炳.变结构控制的理论及设计方法[M].北京:科学出版社,1996
[22]Yang T.Impulsive control theory(J].Lecture Notes in Control and Information Sciences,2001,72:278-284
[23]Zhang Q J,LU J A.Impulsive control and synchronization of a critical chaotic system(J]Wuhan University Journal of Natural Sciences,2007,12(3):426-430
[24]Yang T,Yang L,Yang C.Impulsive control of Lorenz system(J].Physics Letters D,1997110:18-24
[25]Chen L,Chen G R.Fuzzy predictive control of uncertain chaotic systems using time series[J].International Journal of Bifurcation and Chaos,1999,9(4):757-767
[26]Oscar.Fuzzy control of chaos[J].International Journal of Bifurcation and Chaos,1998,8(8):1743-1747
[27]Alsing P M,Garielides A.Using neural networks for controlling chaos[J].Physica Review Letter,1994,49:1225-1231
[28]Otawara K,Fan L T.Controlling chaos with an artificial neural networks[J].Proceedings of the 2nd International IEEE Conference Fuzzy Systems,1995,4:943-948
[29]Lin C T.Controlling chaos by GA based reinforcement learning neural network[J].IEEE Transactions on Neural Networks,1999,10:846-859
[30]薛月菊,冯汝鹏.连续时间耦合系统中时空混沌的自适应模糊控制[J].物理学报,2001,50(3):440-444
[31]裴文江,黄俊,刘文波,于盛林.自适应延迟反馈控制混沌[J].控制理论与应用,1999,16(2):297-300
[32]Matveev A S,Savkin A V.Qualitative theory of hybrid dynamical systems[M].Boston:Birkhauser,2000
[33]高春华,王慧,李平.混合系统建模分析与综合:研究进展与展望[J].系统工程理论与实践,2002,11:15-20
[34]方敏,张雅顺,李辉.混合系统的形式验证方法[J].系统仿真学报,2006,18(10):28-32
[35]Perkins I R,Kumar P R.Stable distributed real-time scheduling of flexible manufacturing/assembly/disassembly systems[J].IEEE Transactions Control,1989,34:139-148
[36]Horn C,Ramadge P J.Dynamics of switched arrival systems with thresholds[J].Proceedings of 32th Conference on Decision and Control,1993,288-293
[37]Chase C,Serrano J,Ramadge P J.Periodicity and chaos from switched flow systems:contrasting examples of discretely controlled continuous systems[J].IEEE Transactions on Automatic Control,1993,38(1):70-83
[38]Ushio T,Ueda H,Hirai K.Controlling chaos in a switched arrival system[J].System Control Letter,1995,26:335-339
[39]Ushio T,Ueda H,Hirai K.Stabilization of periodic orbits in switched arrival systems with N buffers[J].Proceedings of 36th Conference on Decision and Control,1996,1213-1214
[40]Li Wei,Ushio T.Control of a chaotic switched arrival system with controlled internal connections[J].International Journal of Bifurcation and Chaos,2006,16(3):701-707
[41]Tian Y P.Detecting unstable period orbit in switched arrival system[J].Proceedings of the 42nd IEEE Conference on Decision and Control,2003,11:1884-1888
[42]Tian Y P,Yu X H,Chua L O.Time delayed impulsive control of chaotic hybrid systems [J].International Journal of Bifurcation and Chaos,2004,14:1091-1104
[43]Tian Y P.Delayed feedback control of chaos in a switched arrival system[J].Physics Letters A,2005,339(6):446-454
[44]Horn C,Ramadge P J.A topological analysis of a family of dynamical systems with nonstandard chaotic and periodic behavior[J].International Journal of Control,1997,67:979-996
[45]李庆扬,关治,白峰杉.数值计算原理[M].北京:清华大学出版社,2000
[46]Katzorke I,Pikovsky A.Chaos and complexity in simple models of production dynamics [J].Discrete Dynamical Systems,2000,5:179-187
[47]Rem B,Armbruster D.Control and synchronization in switched arrival systems[J].Chaos,2003,13(1):1054-1500
[2]Ruelle D,Takens F.On the nature of turbulene[J].Commun Math Physics,1971,20:167-192
[3]Ruelle D,Takens F.On the nature of turbulene[J].Commun Math Physics,1971,23:343-344
[4]May R M.Simple mathematical models with very complicated dynamics(J].Nature,1976261(5560):459-467
[5]Feigenbaum M J.Quantitative universality for a class of nonlinear transformation(J].Statistical Physics,1978,19:25-52
[6]Guckenheimer J,Holmes P J.Nonlinear oscillations,dynamical systems,and bifurcations of vector fields(M].Springer-Verlag:New York,Berlin,Heidelberg,1983
[7]刘秉正.非线性动力学与混沌基础[M].长春:东北师范大学出版社,1994
[8]Wiggins S.Introduction to applied nonlinear dynamical systems and chaos(M].Springer-Verlag:New York,Berlin,Heidelberg,1990
[9]马知恩,周义仓.常微分方程定性与稳定性方法[M].北京:科学出版社,2005
[10]张琪昌,王洪礼,竺致文,等.分又与混沌理论与应用[M].天津:天津大学出版,2005
[11]关新平,范正平,陈彩莲,华长春.混沌控制及其在保密通信中的应用[M].北京:国防工业出版社,2002
[12]Ott E,Grebogi C,Yorke J A.Controlling chaos(J].Physica Review Letter,1990,64(11):1196-1190
[13]杨凌,刘曾荣.OGY方法的改进及证明[J].应用数学和力学,1998,19(1):127-130
[14]Romeiras F J,Grebogi C,Ott E,Dayawansa W P.Controlling chaotic dynamical systems [J].Physics Letters D,1992,58:165-169
[15]Pyragas K.Continuous control of chaos by self-controlling feedback(J].Physics Letters A,1992,170:421-428
[16]Pyragas K.Experimental control of chaos by delayed self-controlling feedback(J].Physics Letters A,1993,180:99-102
[17]Huberman B A.Dynamics of adaptive systems(J].IEEE Transactions on Circuits and Systems,1990,37(4):547-550
[18]Sinha S.Adaptive control in nonlinear dynamics.Physics Letters D,1990,43:118-128
[19]Utkin V I.Sliding modes and their application in variable structare systems(M].Moscow,Russia:Mir Publishers,1978
[20]Utkin V 1.Variables structure systems:present and future(J].Automation and Remote Control,1983,44(9):1105-1120
[21]高为炳.变结构控制的理论及设计方法[M].北京:科学出版社,1996
[22]Yang T.Impulsive control theory(J].Lecture Notes in Control and Information Sciences,2001,72:278-284
[23]Zhang Q J,LU J A.Impulsive control and synchronization of a critical chaotic system(J]Wuhan University Journal of Natural Sciences,2007,12(3):426-430
[24]Yang T,Yang L,Yang C.Impulsive control of Lorenz system(J].Physics Letters D,1997110:18-24
[25]Chen L,Chen G R.Fuzzy predictive control of uncertain chaotic systems using time series[J].International Journal of Bifurcation and Chaos,1999,9(4):757-767
[26]Oscar.Fuzzy control of chaos[J].International Journal of Bifurcation and Chaos,1998,8(8):1743-1747
[27]Alsing P M,Garielides A.Using neural networks for controlling chaos[J].Physica Review Letter,1994,49:1225-1231
[28]Otawara K,Fan L T.Controlling chaos with an artificial neural networks[J].Proceedings of the 2nd International IEEE Conference Fuzzy Systems,1995,4:943-948
[29]Lin C T.Controlling chaos by GA based reinforcement learning neural network[J].IEEE Transactions on Neural Networks,1999,10:846-859
[30]薛月菊,冯汝鹏.连续时间耦合系统中时空混沌的自适应模糊控制[J].物理学报,2001,50(3):440-444
[31]裴文江,黄俊,刘文波,于盛林.自适应延迟反馈控制混沌[J].控制理论与应用,1999,16(2):297-300
[32]Matveev A S,Savkin A V.Qualitative theory of hybrid dynamical systems[M].Boston:Birkhauser,2000
[33]高春华,王慧,李平.混合系统建模分析与综合:研究进展与展望[J].系统工程理论与实践,2002,11:15-20
[34]方敏,张雅顺,李辉.混合系统的形式验证方法[J].系统仿真学报,2006,18(10):28-32
[35]Perkins I R,Kumar P R.Stable distributed real-time scheduling of flexible manufacturing/assembly/disassembly systems[J].IEEE Transactions Control,1989,34:139-148
[36]Horn C,Ramadge P J.Dynamics of switched arrival systems with thresholds[J].Proceedings of 32th Conference on Decision and Control,1993,288-293
[37]Chase C,Serrano J,Ramadge P J.Periodicity and chaos from switched flow systems:contrasting examples of discretely controlled continuous systems[J].IEEE Transactions on Automatic Control,1993,38(1):70-83
[38]Ushio T,Ueda H,Hirai K.Controlling chaos in a switched arrival system[J].System Control Letter,1995,26:335-339
[39]Ushio T,Ueda H,Hirai K.Stabilization of periodic orbits in switched arrival systems with N buffers[J].Proceedings of 36th Conference on Decision and Control,1996,1213-1214
[40]Li Wei,Ushio T.Control of a chaotic switched arrival system with controlled internal connections[J].International Journal of Bifurcation and Chaos,2006,16(3):701-707
[41]Tian Y P.Detecting unstable period orbit in switched arrival system[J].Proceedings of the 42nd IEEE Conference on Decision and Control,2003,11:1884-1888
[42]Tian Y P,Yu X H,Chua L O.Time delayed impulsive control of chaotic hybrid systems [J].International Journal of Bifurcation and Chaos,2004,14:1091-1104
[43]Tian Y P.Delayed feedback control of chaos in a switched arrival system[J].Physics Letters A,2005,339(6):446-454
[44]Horn C,Ramadge P J.A topological analysis of a family of dynamical systems with nonstandard chaotic and periodic behavior[J].International Journal of Control,1997,67:979-996
[45]李庆扬,关治,白峰杉.数值计算原理[M].北京:清华大学出版社,2000
[46]Katzorke I,Pikovsky A.Chaos and complexity in simple models of production dynamics [J].Discrete Dynamical Systems,2000,5:179-187
[47]Rem B,Armbruster D.Control and synchronization in switched arrival systems[J].Chaos,2003,13(1):1054-1500